1. ECE 3110: Introduction to Digital Systems Review #1 (Chapter 1,2)

2. 2 Chapters 1,2 summary Analog Digital: advantages, electronic aspects, software aspects, digital design levels. IC: wafer, die, classifications, PLC,ASIC Positional number systems: binary, octal, decimal,hex Unsigned numbers: conversions, addition/subtraction/multiplication/division Signed numbers: representations, conversions, addition/subtraction Sign extension Overflow Binary Codes for decimal numbers:BCD,2421,excess-3…. Gray code Other codes

3. 3 Digital Design Basics Analog vs. Digital Why we need digital?  Reproducibility, economy, programmability… Digital Devices  Gates, FFs  Combinational, sequential circuits

4. 4 Electronic and software aspect of digital design Digital abstraction Noise Margin specification Examples of software tools

5. 5 Integrated Circuits (IC) A collection of one or more gates fabricated on a single silicon chip.  Wafer, die Small-scale integration (SSI): 1-20  DIP: dual in-line-pin package  Pin diagram, pinout MSI: 20-200 gates LSI: 200-200,000  VLSI: >100,000, 50million (1999)

6. 6 Programmable Logic Device (PLD) ICs which Logic function can be programmed after manufacture. 2-level AND-OR gates using user-programmable connections  PLAs: programmable Logic Arrarys.  PALs: programmable array logic devices  Programmable logic devices (PLDs) CPLD: complex PLD FPGA: Field-Programmable Gate Array

7. 7 Application-Specific ICs (ASIC) Semi-custom IC: chips designed for a particular, limited product/application Reducing chip count, size, power consumption Higher performance. NRE: nonrecurring engineering cost  Standard cells: library  Gate array: an IC with internal structure as an array of gates, unspecified interconnection

8. 8 Digital Design Levels Many representations of digital logic Device Physics and IC manufacturing  Moore’s Law [1965, Gordon Moore]: Transistor level --->Logic design, functional building blocks Transistor level, Truth table, gate level logic, prepackaged blocks, equations, HDL

9. 9 Binary Representation The basis of all digital data is binary representation. Binary - means ‘two’  1, 0  True, False  Hot, Cold  On, Off We must be able to handle more than just values for real world problems  1, 0, 56  True, False, Maybe  Hot, Cold, Warm, Cool  On, Off, Leaky

10. 10 Positional Notation Base Radix Weight Radix 2, 8, 10,16

11. 11 Unsigned numbers N binary digits (N bits) can represent unsigned integers from 0 to 2 N -1. Conversions:  Hex binary  Octal binary (padded with zero)  Any base decimal Operations (binary): addition/subtraction/multiplication/division

12. 12 Hex to Binary, Binary to Hex A2F 16 = 1010 0010 1111 2 345 16 = 0011 0100 0101 2 Binary to Hex is just the opposite, create groups of 4 bits starting with least significant bits. If last group does not have 4 bits, then pad with zeros for unsigned numbers. 1010001 2 = 0101 0001 2 = 51 16 Padded with a zero

13. 13 Conversion of Any Base to Decimal Converting from ANY base to decimal is done by multiplying each digit by its weight and summing. 1011.11 2 = 1x2 3 + 0x2 2 + 1x2 1 + 1x2 0 + 1x2 -1 + 1x2 -2 = 8 + 0 + 2 + 1 + 0.5 + 0.25 = 11.75 Binary to Decimal Hex to Decimal A2F 16 = 10x16 2 + 2x16 1 + 15x16 0 = 10 x 256 + 2 x 16 + 15 x 1 = 2560 + 32 + 15 = 2607

14. 14 Conversion of Decimal Integer To ANY Base Divide Number N by base R until quotient is 0. Remainder at EACH step is a digit in base R, from Least Significant digit to Most significant digit.

15. 15 Conversion of Decimal Integer To ANY Base Example Convert 53 to binary 53/2 = 26, rem = 1 26/2 = 13, rem = 0 13/2 = 6, rem = 1 6 /2 = 3, rem = 0 3/2 = 1, rem = 1 1/2 = 0, rem = 1 53 10 = 110101 2 = 1x2 5 + 1x2 4 + 0x2 3 + 1x2 2 + 0x2 1 + 1x2 0 = 32 + 16 + 0 + 4 + 0 + 1 = 53 Least Significant Digit Most Significant Digit

16. 16 c in or b in XYC out sB out d 0000000 0010111 0100101 0111000 1000111 1011010 1101000 1111111 Binary addition/subtraction rules

17. 17 Binary, Decimal addition 34 + 17 ------ 51 from LSD to MSD: 7+4 = 1; with carry out of 1 to next column 1 (carry) + 3 + 1 = 5. answer = 51. Decimal 101011 2 + 000001 2 --------------- 101100 2 From LSB to MSB: 1+1 = 0, carry of 1 1 (carry)+1+0 = 0, carry of 1 1 (carry)+0 + 0 = 1, no carry 1 +0 = 1 0 + 0 = 0 1 + 0 = 1 answer = 101100 2 Binary

18. 18 Subtraction Decimal 900 - 001 ------- 899 0-1 = 9; with borrow of 1 from next column 0 -1 (borrow) - 0 = 9, with borrow of 1 9 - 1 (borrow) - 0 = 8. Answer = 899. Binary 100 2 - 001 2 ------- 011 2 0-1 = 1; with borrow of 1 from next column 0 -1 (borrow) - 0 = 1, with borrow of 1 1 - 1 (borrow) - 0 = 0. Answer = 011 2.

19. 19 Representation of Negative Numbers Signed-Magnitude Representation: Negates a number by changing its sign. Complement Number Systems: negates a number by taking its complement.  Diminished Radix-Complement Representation One’s-Complement  Radix-Complement Representation Two’s-Complement

20. 20 NOTE:  Fix number of digits  SM, 1’s complement, 2’s complement may be different for NEGATIVE numbers, but  for positive numbers, the representations in SM, 1’s complement, 2’s complement are the SAME, equals to the unsigned binary representation.

21. 21 Ranges (N bits) 1’s complement can represent the signed integers -(2 (N-1) - 1) to + (2 (N-1) - 1 ) unsigned binary can represent unsigned integers from 0 to 2 N -1. SM can represent the signed integers -(2 (N-1) - 1) to + (2 (N-1) - 1 ) 2’s complement can represent the signed integers -2 (N-1) to + (2 (N-1) - 1)

22. 22 Sign extension For unsigned binary, Just add zeros to the left. For signed binary (SM,1’s,2’s complement…):  Take whatever the SIGN BIT is, and extend it to the left.

23. 23 Conversions for signed numbers Hex--->signed decimal Given a Hex number, and you are told to convert to a signed integer (either as signed magnitude, 1s complement, 2s complement)  Step 1: Determine the sign  Step 2: determine magnitude  Step 3: combine sign and magnitude Signed decimal ---->hex  Step 1: Know what format you are converting to!!!  Step 2: Ignore the sign, convert the magnitude of the number to binary.  Step 3 (positive decimal number): If the decimal number was positive, then you are finished no matter what the format is!  Step 3 (negative decimal number): more work need to do.

24. 24 Hex to Signed Decimal Conversion Rules STEP 1: Determine the sign! If the Most Significant Bit is zero, the sign is positive. If the MSB is one, the sign is negative. This is true for ALL THREE representations: SM, 1s complement, 2s complement. STEP 2 (positive sign): If the sign is POSITIVE, then just convert the hex value to decimal. The representation is the same for SM, 1s complement, 2s complement. STEP 2 (negative sign): If the sign is Negative, then need to compute the magnitude of the number. If the number is SM format, set Sign bit to Zero If the number is 1s complement, complement each bit. If the number is 2s complement, complement and add one. STEP 3 : Just combine the sign and magnitude to get the result.

25. 25 Signed Decimal to Hex conversion Step 1: Know what format you are converting to!!! You must know if you are converting the signed decimal to SM, 1s complement, or 2s complement. Step 2: Ignore the sign, convert the magnitude of the number to binary. Step 3 (positive decimal number): If the decimal number was positive, then you are finished no matter what the format is! Step 3 (negative decimal number): Need to do more if decimal number was negative. If converting to SM format, set Sign bit to One: If converting to 1s complement, complement each bit. If converting to 2s complement, complement each bit and add 1.

26. 26 signed addition/subtraction Two’s-complement  Addition rules  Subtraction rules Overflow:  Out of range  Detecting unsigned overflow (carry out of MSB)  Detecting 2’s complement overflow

27. 27 Detecting Two’s Complement Overflow Two’s complement overflow occurs is: Add two POSITIVE numbers and get a NEGATIVE result Add two NEGATIVE numbers and get a POSITIVE result We CANNOT get two’s complement overflow if I add a NEGATIVE and a POSITIVE number together. The Carry out of the Most Significant Bit means nothing if the numbers are two’s complement numbers.

28. 28 Codes Code: A set of n-bit strings in which different bit strings represent different numbers or other things. Code word: a particular combination of n-bit values  N-bit strings at most contain 2 n valid code words. To represent 10 decimal digits, at least need 4 bits. Excessive ways to choose ten 4-bit words. Some common codes:  BCD: Binary-coded decimal, also known as 8421 code  Excess-3  2421… Codes can be used to represent numerical numbers, nonnumeric texts, events/actions/states/conditions

29. 29 How to construct Gray Code Recursively  A 1-bit Gray Code has 2 code words, 0, 1  The first 2 n code words of an (n+1)-bit Gray code equal the code words of an n-bit Gray Code, written in order with a leading 0 appended.  The last 2 n code words equal the code words of an n-bit Gray Code, but written in reverse order with a leading 1 appended.

30. 30 Another method to construct Gray Code The bits of an n-bit binary or Gray-code word are numbered from right to left, from 0 to n-1 Bit i of a Gray code word is  0 if bits i and i+1 of the corresponding binary code words are the same  1: otherwise